# A Mathematical Universe

In Michio Kaku’s book “hyperspace: a scientific odyssey through parallel universes, time warps, and the 10th dimension,” Kaku describes a moment that inspired his intellectual journey. “When I was 8 years old, I heard a story that would stay with me for the rest of my life. I remember my schoolteachers telling the class about a great scientist who had just died. They talked about him with great reverence, calling him one of the greatest scientists in all history… I didn’t understand much of what they were trying to tell us, but what most intrigued me about this man was that he died before he could complete his greatest discovery. They said he spent years on this theory, but he died with his unfinished papers still sitting on his desk.” Kaku credits this mystery as contributing to his desire to pursue physics and a deeper understanding of the world. The man in the story was Albert Einstein, and the theory was a unified theory of physics.

Einstein is a household name in physics for good reason. Through the simple act of asking questions, and exploring the logical ramifications of those questions no matter how unusual, Einstein was able to reason his way into a new theory of gravity: relativity. The problem was that relativity, regardless of how successful it was as a theory, only explained gravity; the other fundamental forces, electromagnetism and the nuclear forces, were not addressed. Einstein’s final task, which he never completed, was to unify gravity with these other fundamental forces. To create a theory that accounted for all the fundamental forces in physics. However, Einstein did not succeed, and the search for such a unifying “theory of everything” inspired a generation of physicists, Kaku among them.

## Thinking in higher dimensions

We humans exist in a three-dimensional world. Objects have height, width and breadth, and to identify an object’s location on our planet, we would need to identify three numbers: a latitude, a longitude, and an altitude. We could think of time existing as a fourth dimension, duration, but we can only do that if we accept that it is a different type of dimension as we experience it differently. I can rotate an object in three spatial dimensions, but I can’t rotate something in time. The dominant theory of time prior to Einstein’s relativity was Newton’s mechanics. Newton viewed time as an immutable quantity. Time moved forward at the same pace regardless of who, or what, was measuring it. To Newton, time was a universal constant that, unlike space, could not be changed. Relativity changed this. In relativity, time isn’t fixed but can bend. Einstein’s theory of special relativity postulates that the experience of time is ‘relative’ to how fast an observer is moving. As my speed, compared to an observer, increases both our experiences of time and space also change. Time for fast-moving travellers will be observed to be passing slower to their stationary friends; this is known as time dilation. As well, the perceived size of a fast-moving traveller will appear to contract: length contraction. General relativity takes this idea one step further by recognizing that our experience of acceleration and our experience of gravity is fundamentally the same thing. Large masses bend both space and time in a similar fashion. Relativity insists that space and time are not two separate entities that follow two separate sets of rules. Instead, they are a single object, spacetime, following a unified set of rules. Thus, Einstein simplified physics by adding a higher dimension

Kaku’s book introduces this idea from Einstein and follows it through a number of logical expansions. If adding a fourth dimension allows us to explain gravity through geometry, then maybe we can add even more dimensions to help us to explain the other fundamental forces. Hyperspace is ultimately a book about string theory, and all the various false starts and dead ends physicists took to expand Einstein’s four dimensions into the ten that the theory requires. Fundamentally, Kaku is arguing that the laws of physics simplify when viewed from higher dimensions.

Hyperspace was an important early inspiration for my own intellectual journey. The book was my first introduction to physical theory, and Kaku’s main argument has stuck with me to this day. Of course, I was a child when I first read it, and my underdeveloped brain didn’t understand anything about the physics or mathematics Kaku was arguing for. Instead, I connected to the simpler explanations of how higher dimensions can make possible the seemingly impossible.

Imagine a two-dimensional creature whose entire world is a single piece of paper. From the perspective of this creature, we humans can do the impossible. We can bend the paper in on itself, causing two ends to touch and allowing the creature to instantly ‘warp’ from one edge of their universe to another. We could also remove this creature from their paper world, ‘turn them over’, and magically transform right into left. Through a simple act of geometry, we can permanently disfigure the creature because it is unable to flip itself back due to that act requiring a third dimension. Likewise, if two such creatures saw each other, they would only be able to describe the exterior shell or outline of each other, but we can easily describe their internals by looking at them from above the page. This knowledge is trivially gained by us but is functionally unknowable to the two dimensional creatures.

In a single chapter Kaku taught me a single unknowable truth, how to visualize a four dimensional cube, and in doing so convinced me that there is no reason why I couldn’t be made to understand the unknowable truth of the universe as well; even those parts I couldn’t experience for myself.

It seems almost trivial now, but the simple idea of higher dimensions beyond the three that we live in changed everything for me. It opened up the possibility of two simple truths. The first is that there are things in this world that I am physically incapable of experiencing, like extra dimensions, that exist, and effects me. The second is that even though such a reality is physically beyond me, I can still come to understand it. In a single chapter Kaku taught me a single unknowable truth, how to visualize a four dimensional cube, and in doing so convinced me that there is no reason why I couldn’t be made to understand the unknowable truth of the universe as well; even those parts I couldn’t experience for myself.

Yet, Kaku’s higher dimensions are in themselves a bit of a trap. Even though Einstein unified the three dimensions into four dimensions, time still stands alone. Imagine the second hand on the face of a clock. When the hand points to the twelve it points straight up and all of its length is along the vertical dimension. As time passes, and the hand rotates, the length becomes less vertical and more horizontal. Slowly all of its vertical orientation will be transformed into a horizontal orientation. Rotation in space merely transforms vertical orientation into horizontal orientation, an objects true length never gets bigger. Time works opposite to this. As an object speeds up, other observers will notice that time for them appears to slow down thus growing larger. As an object approaches the speed of light, its maximum rotation, it will appear to have an infinite length. The shortest possible measurement of time will always come from the person who is experiencing it, everyone else will measure something larger with no upper bound. Thus rotation in time can expand sizes from their true size all the way to infinity. Time is still special because it works backwards to space.

The equations of relativity are written in a mathematical framework called ‘Riemann geometry’ which is a system for expressing a multitude of such exotic, or ‘non-spacial’, geometries using the same common language. Both space and time fit within this paradigm and thus are equally expressible as a four dimensional Riemann manifold. However, ‘dimension’ in this context suddenly becomes a distinctly mathematical term and losses a lot of context that the English word ‘dimension’ might imply. When Kaku says that the laws of physics simplify at higher dimensions, he is referring to a distinctly mathematical definition of the term ‘simple’ that merely implies that these two different things can be expressed using the same terminology. Time and space still follow different rules as they are still different, the formula that describes spacetime as a single unified entity still needs to place a negative sign in front of the temporal coordinate to set it apart from the other spacial dimensions. Unfortunately, it is all too easy to jump to the conclusion that just because some things can be expressed within the same framework all things eventually will be as well. String theory begins with this assumption, and twenty years later it has yet to produce anything from this claim. If the other fundamental forces are just an extension of spacetime, then we have yet to discover what symbols need to be placed in front of them to make them work.

## Pythagoras

Pythagoras was an ancient Greek philosopher whose name will be familiar to anyone who has studied dimensions. The very formula that describes rotation in a spatial dimension carries his name. The Pythagorean theorem states that the three sides of a right-angle triangle are related. That the square of the hypotenuse, the side opposite the right angle, is equal to the sum of the squares on both additional sides: A^2 + B^2 = c^2. As the second hand rotates its experience of the vertical and horizontal dimensions changes according to that formula. If the vertical dimension says that the hand is three centimetres long and the horizontal says it is four then the Pythagorean theorem tells us that its true length is five centimetres. Unfortunately, apart from this theorem very little in modern mathematics is attributed to that man. In fact, very little about Pythagoras’s actual beliefs and teachings are known or even can be known for certain today. This is because he wrote nothing down and everything we do know comes to us through other sources; most of which were written hundreds or even thousands of years after his death. Worse, even his theorem likely did not come from him. Modern archaeology supplies plenty of evidence that the Pythagorean theorem was known in some form or another in ancient Egypt, a place where folklore states Pythagoras studied. At best he only rediscovered or popularized the theorem. At worst had nothing to do with it. However, what we do know is that Pythagoras, or the Pythagoreans, believed that deep down at its core the universe was made out of numbers.

Legends have it that Pythagoras also had a Eureka moment that formed the foundation of his view of the universe. Kitty Furgeson in her book, “Life of Pythagoras” describes this moment as such. While experimenting with the strings of a lyre Pythagoras, “(or someone inspired by Pythagoras) discovered that the connections between lyre string length and the human ears are not arbitrary or accidental. The ratios that underlie musical harmony make sense in a remarkably simple way. In a flash of extraordinary clarity, the Pythagorean found that there is pattern and order hidden behind the apparent variety and confusion of nature, and that it is possible to understand it through numbers. ” The details of this statement are of course up for debate. Iamblicus, an important early biographer of Pythagoras, claims that Pythagoras came to this understanding while listening to the sounds of a hammer striking an anvil not while playing with a lyre. As well, the modern reader won’t find much enlightenment in what fragments we do have of Pythagoras’ metaphysics. Most of it sounds like numerology. The Pythagoreans believed in a theory of music that governed the heavens. The strings on a Lyre could be tuned in an unlimited number of ways, but it was only when they were tuned to the integer ratios that it could generate harmony, the same is true for the heavens. Pythagoras believed that each celestial body orbited a “central fire” and produced a sound as it travelled. Each heavenly body produced a different sound and together produced harmony. Ten was an important number to Pythagoras because it represented perfection in his system. One can create a perfect triangle by starting with a base of four round balls and stacking three, then two, and finally a single ball on top of it for a total of ten balls. Likewise, Pythagoras needed for there to be exactly ten celestial objects orbiting the ‘inner fire’: the sun, the moon, the earth, five planets, the stars, and a mysterious counter earth that was never visible because it was always hidden on the other side of the inner fire. This counter earth existed not because he had observed it, but because his model wouldn’t make sense without it.

The details of what Pythagoras actually believed are eternally up for debate; we can’t really know anything for certain. But his impact was undoubtedly profound. Plato, inspired by his conversations with Pythagorean followers, included a detailed geometric view of the cosmos in his Timaeus. In his model, the world was literally geometry and made up of atomic triangles. Each of the four basic elements, water, earth, fire, and air, gain their properties through the configuration of the triangles within them, and the world as a whole comes out of the interaction of these elements. Modern readers might see this construction as nonsense, but if we remember that the foundation of modern mathematics, Euclid’s elements, had not yet been written it’s much easier to see that the geometric view of the universe in Plato’s Timaeus is at least an attempt to explore the ramifications of a mathematical universe. As mathematics has developed so too have the mathematical models that describe our universe. There is no shortage of such models we could explore. Some are absurd, like Kepler’s early model of the solar system which used platonic solids to describe the orbits of the planets, some are useful, like Brahe’s geocentric model which did accurately predict how certain planets moved but was ultimately proven to be false, and some would fundamentally alter how generations of scientists view the universe, like Einstein’s relativity. So the question remains, is there a correct fundamental model of the universe?

## Conclusion

Physicist Max Tegmark takes this idea to its ultimate extreme in his book, “Our Mathematical Universe” where he argues that the universe isn’t just described by a mathematical model, it is a mathematical object. He calls this the Mathematical Universe Hypothesis. “If the Mathematical Universe Hypothesis is correct, then our Universe is a mathematical structure, and from its description, an infinitly intelligent mathematician should be able to derive all these physical theories.”

Fundamentally, this is the question that stuck with me into adulthood. Does such a theory exist? I don’t mean does a unified theory of the fundamental physical forces exist? For all I know, Tegmark’s confidence that we will be printing t-shirts with the equations of a unified theory of physics in our lifetime could be correct, but that question doesn’t interest me. Instead, what I’m asking is something more profound. Is there a unified theory of everything? Can a sufficiently powerful, infinite-dimensional, creature derive our universe, everything inside it, and everything it is capable of becoming from a single framework. Thinking back to my childhood both myself and Pythagoras had a similar moment. Reading Kaku gave me a brief moment of enlightenment where I realized that part of the world isn’t random. However, unlike Pythagoras, my religious background prevented me from taking the next step. Just because something is understandable doesn’t imply that everything is. Just because two fundamental forces can be unified in geometry doesn’t mean that all of them can either.

The problem with the mathematical universe hypothesis and the search for a “unified theory of everything” is that neither of them is falsifiable. Sure Pythagorus’ model containing exactly ten celestial bodies is wrong, Newton’s theory describing a static and universal time is wrong, Einstein’s failure to account for electromagnetism and the nuclear forces implies that his theory is at best incomplete, and string theory’s inability to predict any experimental outcome implies that it probably isn’t the final theory either. Yet, none of these precludes the idea that a correct and perfect mathematical model of the universe doesn’t exist. There’s just no way of proving a negative. Worse, the fact that each of these models improves on the previous model implies some sort of forward momentum. Each scientific breakthrough doesn’t invalidate the knowledge gained from the previous model, it merely casts that knowledge from a new perspective, a higher dimension, that lets us see old knowledge along with new knowledge in a common framework. So how do we account for the success of the mathematical sciences without admitting that at some level the universe is mathematical?

To begin searching for knowledge we must first admit that such a thing, a stable foundation, exists at all. If the universe exists then it must be true to its own nature. If the universe changes it is not because the nature of the universe has changed, it is because the nature of the universe is to change.

What is mathematics if not the language of structure? If I say that 1 + 1 = 2, I am asserting a relationship between these two objects that offer clues as to the structure these objects live in. If this statement is only true sometimes, or in some contexts, then that implies that the rules governing these objects still dictate that 1 + 1 = 2 in such contexts. Thus our knowledge might not be universal, but it still hints at a fundamental nature we still know something about. To begin searching for knowledge we must first admit that such a thing, a stable foundation, exists at all. If the universe exists then it must be true to its own nature. If the universe changes it is not because the nature of the universe has changed, it is because the nature of the universe is to change. If we assume that the universe is true to its own nature and that it follows its own rules then mathematics inevitably leaks out.

At this juncture, we have not built a foundation strong enough to even explore that assumption in full. However, what we can do is explore what it means for something to follow its own rules and to be mathematical. This is where my own personal journey began. If mathematics can generate truth, then what does a mathematical universe look like? What, if anything, can maths tell us about itself.